Method For Controlling The Actuator Of The Wastegate Of A Turbocharger Of A Motor Vehicle

ABSTRACT

The disclosure relates to internal combustion engines. The teachings thereof may be embodied in methods for controlling the actuator of the wastegate of an exhaust gas turbocharger of a motor vehicle. A method for controlling an actuator of the wastegate of an exhaust gas turbocharger of a motor vehicle may include: characterizing the wastegate in a model as a series connection of two throttle points; and actuating the wastegate based on the model.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a U.S. National Stage Application of InternationalApplication No. PCT/EP2015/067991 filed Aug. 4, 2015, which designatesthe United States of America, and claims priority to DE Application No.10 2014 217 456.2 filed Sep. 2, 2014, the contents of which are herebyincorporated by reference in their entirety.

TECHNICAL FIELD

The disclosure relates to internal combustion engines. The teachingsthereof may be embodied in methods for controlling the actuator of thewastegate of an exhaust gas turbocharger of a motor vehicle.

BACKGROUND

In combustion engines with turbocharging, the fresh air is compressed bymeans of a turbocharger before flowing into the cylinders in order tointroduce a larger air mass into the cylinder than is possible bysuction from the respective ambient pressure. The resulting chargingpressure p₂, that is, the pressure after turbocharger compressor, andthe air mass flow through the turbocharger compressor are determined bythe combination of turbocharger speed and turbocharger power.

SUMMARY

The present disclosure teaches at least a wastegate model, which,depending on the respective application, is used directly or inverted asan algorithm for controlling the turbocharger, as is explained ingreater detail below. The teachings may be embodied in methods forcontrolling the actuator of the wastegate of an exhaust gas turbochargerof a motor vehicle, characterized in that the control signal isdetermined by taking into consideration a model, which describes thewastegate as a series connection of two throttle points.

In some embodiments, a characteristic diagram is filed in a memory ofthe engine control device of the motor vehicle and describes the nominalrelationship of annular surface to borehole surface of the wastegate asa function of the pressure relationships at the wastegate and as afunction of a nominal mass flow factor.

Some embodiments may include: determining the nominal wastegate exhaustgas mass flow (m_(wg,sp)) at a current operating point during therunning time of the exhaust gas turbocharger, determining a nominal massflow factor (W_(sp)) belonging to the current operating point by usingthe determined nominal wastegate exhaust gas mass flow, determining anominal wastegate-area relationship (Q_(A,sp)), belonging to the currentoperating point, from the filed characteristic diagram by using thedetermined nominal mass flow factor, and determining the nominalposition (s_(acr,sp)) of the actuator realizing the required nominalwastegate mass flow in the current operating point.

Some embodiments may include: determining a nominal force (F_(p,sp)) onthe wastegate plate of the wastegate, determining a nominal actuatorpressure (P_(acr,sp)), required for setting a desired charging pressure,from the determined nominal position (S_(acr,sp)) and the determinednominal force (F_(p,sp)), and determining the control signal (u_(wg))for the actuator by using the nominal actuator pressure determined.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 shows a functional sketch of a wastegate actuator which may beused to implement the teachings of the present disclosure;

FIG. 2 shows a functional sketch of an electropneumatic wastegateactuator used to implement the teachings of the present disclosure;

FIG. 3 shows a detail sketch of the wastegate;

FIG. 4 shows a diagrammatic representation of a wastegate as a seriesconnection of two throttle points,

FIG. 5 shows a sketch of the course of the flow coefficients as afunction of the pressure relationship at a throttle point,

FIG. 6 shows a sketch to illustrate the course of the flow coefficientsand second substitute functions as a function of the pressurerelationship at a throttle point,

FIG. 7 shows a three-dimensional sketch of the course of the substitutefunction φ(π_(R), π_(B)),

FIG. 8 shows a three-dimensional sketch of the course of the substitutefunction ψ(π_(R), π_(B)),

FIG. 9 shows a three-dimensional sketch to illustrate a graphic solutionof an equation,

FIG. 10 shows a three-dimensional sketch to illustrate the course of theglobal stationary pressure relationships over the annular surface of thewastegate as a function of the ratio of pressure before and after theturbine and a wastegate surface ratio and

FIG. 11 shows a three-dimensional sketch to illustrate the course of themass flow factor as a function of the wastegate area relationship andthe ratio of the pressure upstream and downstream from the turbine.

DETAILED DESCRIPTION

The turbocharger power or turbine output P_(tur) is determined by

$\begin{matrix}{{P_{wg} = {{\overset{.}{m}}_{sp} \cdot {T_{3}\left\lbrack {1 - \left( \frac{p_{4}}{p_{3}} \right)^{\frac{\kappa - 1}{\kappa}}} \right\rbrack} \cdot c_{p} \cdot \eta_{acr}}},} & (1)\end{matrix}$

with {dot over (m)}_(tur)=turbine mass flow, T₃=exhaust gas temperatureupstream from the turbine, p₃=pressure upstream from the turbine,p₄=pressure downstream from the turbine, c_(p)=specific heat capacity ofthe exhaust gas under constant pressure and η_(tur)=turbine efficiency.

For turbochargers with a wastegate, the turbine power—and thus,indirectly, the charging pressure and the engine power—are therebycontrolled in that the exhaust gas mass flow, occurring in therespective operating point of the internal combustion engine, from thecylinders {dot over (m)}_(eng), through a specific opening of thewastegate, which is determined by the wastegate position s_(wg), isdivided into a turbine mass flow {dot over (m)}_(tur), which, at therespective prevailing pressures and temperatures of equation (1), bringsabout the required turbocharger power and a wastegate mass flow {dotover (m)}_(wg), is bypassed at the turbine and does not contribute tothe turbocharger power:

{dot over (m)} _(eng) ={dot over (m)} _(tur) +{dot over (m)} _(wg).  (2)

FIG. 1 shows a functional sketch of a wastegate actuator which may beused to implement the teachings of the present disclosure. In FIG. 1,the following are illustrated: a wastegate bore 2 in the turbine housing1, which is closed off on its right side by a wastegate plate 3, thepressure p₃ upstream from the turbine, the pressure p₄ downstream fromthe turbine, the exhaust gas mass flow {dot over (m)}_(wg) through thewastegate, the force F_(p) acting on the wastegate plate due to thepressure difference at the wastegate plate, a wastegate lever 4, whichis mounted in an axis of rotation Z, and has a wastegate-side arm 4 a ofthe length l_(wg) and an actuator-side arm 4 b of length l_(acr), and awastegate actuator rod 6 in a position s_(acr), on which an actuator 7acts with an actuator force F_(acr). Forces opening the wastegate andmoments are defined as positive.

The wastegate position is controlled via a lever mechanism by awastegate actuator, which is actively controlled by the engine controldevice. It is customary to combine a pre-control of the wastegateactuator, which is calculated on the basis of the desired chargingpressure p_(2,sp), with a charging pressure control to minimize thecharging pressure difference

Δp ₂ =p _(2,sp) −p ₂  (3):

u _(wg) =u _(wg,opl)(p _(2,sp))+u _(wg,cll)(p _(2,sp) −p ₂),  (4)

with u_(wg)=Wastegate control, u_(wg,opl)(p_(2,sp))=Wastegatepre-control and u_(wg,cll)(p_(2,sp)−p₂)=Signal to the charging pressurecontroller output.

Accurate control of the waste gate may provide a rapid and accuraterealization of the required engine torque. If the vibrations excited bythe pulsating exhaust gas mass flow are neglected, the wastegateposition s_(wg) is constant exactly then, that is, the wastegate is in astationary state, if the torques, which act upon the wastegate levermounted to rotate about the wastegate axis Z, add up to 0, that is

Σ(M _(Z))=M _(p) +M _(acr)=0,  (5)

with M_(p)=the torque caused by the pressure difference at the wastegateplate, and M_(acr)=torque caused by the actuator.

In systems with positional measurement of the wastegate actuator, thecontrol of the wastegate for setting this torque equilibrium and thusthe desired charging pressure is realized as a two-stage control with:an external control circuit for setting the desired charging pressure bymeans of a preset of the nominal position of the wastegate actuators_(acr,sp)

s _(acr,sp) =s _(acr,opl)(p _(2,sp))+s _(acr,cll)(p _(2,sp) −p ₂),  (6)

with s_(acr,opl)(p_(2,sp))=the pre-control of the wastegate position ands_(acr,cll)(p_(2,sp)−p₂)=the charging pressure controller output;and an internal control circuit for adjusting the nominal wastegateposition required

u _(wg) =u _(wg,opl)(s _(acr,sp))+u _(wg,cll)(s _(acr,sp) −s_(acr)),  (7)

with u_(wg)=wastegate control u_(wg,opl)(s_(acr,sp))=wastegatepre-control and u_(wg,cll)(s_(acr,sp)−s_(acr))=signal to the positioncontroller output.

In systems without position measurement of the wastegate actuator, theactuator position is not known.

FIG. 2 shows a functional sketch of an electropneumatic wastegateactuator used to implement the teachings of the present disclosure. Thewastegate actuator may include: a wastegate borehole 2 in the turbinehousing 1, closed from the right by the wastegate plate 3; the pressurep₃ upstream from the turbine; the pressure p₄ downstream from theturbine; the exhaust gas mass flow {dot over (m)}_(wg) through thewastegate; the force F_(p) acting on the wastegate plate due to thepressure difference at the wastegate plate; a wastegate lever 4, whichis mounted in the axis of rotation Z and has a wastegate-side arm 4 a ofthe length l_(wg) and an actuator-side arm 4 b of length l_(acr), aswell as a wastegate actuator rod 6 in a position s_(acr), on which theactuator acts with an actuator force F_(acr).

In FIG. 2, an electropneumatic reduced pressure wastegate actuatorwithout flow is shown as an example of the execution of a wastegateactuator without position measurement. As depicted, the actuator mayinclude an electropneumatic 3-way valve 8, which, depending on thecontrol PWM_WG (=u_(wg) in the sense of equation (4)) sets an actuatorpressure p_(acr) between the ambient pressure p₀ and the reducedpressure p_(vac), a pneumatic pressure nozzle 7 with a membrane 7 a ofthe active area A_(acr), the membrane 7 a being connected with theactuator rod 6, two chambers 7 b and 7 c separated by the membrane 7 a.The first actuator chamber 7 b is connected with the ambient pressure p₀and the second actuator chamber 7 c with the control pressure p_(acr),which is separated from the surroundings, here for a reduced pressureactuator with p_(acr)<p₀, as well as an actuator spring 7 d with aspring constant k.

The pressure difference at the membrane 7 a results in the controlpressure acting on the actuator rod

F _(acr) =A _(acr)·(p ₀ −p _(acr)).  (8)

The deformation of the spring, into the actuator position s_(acr),results in the spring force, which acts on the actuator rod

F _(spr) =k·s _(acr) +F _(spr,0)  (9)

with F_(spr,0)=the pretension of the spring at s_(acr)=0.

In the configuration shown in FIG. 2, the magnitude of the spring forceF_(spr) closing the wastegate increases with the increasing actuatorposition s_(acr). With that, the spring constant is negative. Thecontrol force and the spring force add up to the actuator force F_(acr):

$\begin{matrix}\begin{matrix}{F_{acr} = {F_{acr} + F_{acr}}} \\{= {{A_{acr} \cdot \left( {p_{0} - p_{acr}} \right)} + {k \cdot s_{acr}} + F_{{acr},B}}}\end{matrix} & (10)\end{matrix}$

Other embodiments of the electropneumatic wastegate actuator, forexample, with an arrangement of the actuator spring in the other chamberor another switching valve or a subjection of the switching valve toother pressures, may only change the amount and possibly the sign of theforces under consideration. The physical dependencies are the same as inthe embodiment depicted.

FIG. 3 shows a detail sketch of the wastegate. From FIG. 3, the turbinehousing 1 with the wastegate borehole 2 of constant diameter D_(wg) andconstant cross-sectional area can be seen. The following applies:

$\begin{matrix}{A_{B} = {\frac{\pi}{4} \cdot D_{wg}^{2}}} & (11)\end{matrix}$

To the right of the turbine housing 1, the wastegate plate 3, which isat a distance S_(wg) from the stop on the turbine housing, is shown. Inthis case, to simplify matters, it is assumed that the movement of thewastegate plate takes place rectilinearly in the direction of the axisof the wastegate borehole. The following applies:

$\begin{matrix}{s_{wg} = {s_{acr} \cdot \frac{I_{wg}}{I_{acr}}}} & (12)\end{matrix}$

Between the turbine housing 1 and the wastegate plate 3, there is showna cylinder-shaped annular surface, which is envisaged as an extension ofthe wastegate borehole,

$\begin{matrix}{A_{R} = {{\pi \cdot D_{wg} \cdot s_{wg}} = {\pi \cdot \frac{I_{wg}}{I_{acr}} \cdot s_{acr}}}} & (13)\end{matrix}$

through which the wastegate mass flow is discharged after flowingthrough the wastegate borehole. The pressure difference at the wastegateplate exerts a force F_(p) on the wastegate plate and a moment on thewastegate lever:

M _(p) =F _(p) ·l _(wg)  (14).

The actuator force F_(acr), as the sum of the control force F_(ct1) andthe spring force F_(spr) exerts, according to equation (10), a moment onthe wastegate lever of

$\begin{matrix}{\begin{matrix}{M_{acr} = {{F_{acr} \cdot I_{acr}} = {\left( {F_{acr} + F_{spr}} \right) \cdot I_{acr}}}} \\{= {{A_{acr} \cdot \left( {p_{0} - p_{acr}} \right) \cdot I_{acr}} + {\left( {{k \cdot s_{acr}} + F_{{spr},0}} \right) \cdot I_{acr}}}}\end{matrix}.} & (15)\end{matrix}$

By inserting equations (14) and (15) in equation (5), the followingresults:

0=F _(p) −l _(wg) +A _(acr)·(p ₀ −p _(acr))·l _(acr)+(k·s _(acr) +F_(spr,0))·l _(acr)  (16).

The membrane area A_(acr), the lever arm lengths l_(acr), l_(wg), thespring constants k and the spring pre-tension F_(spr,0) are systemconstants. The slowly changing ambient pressure is known in the enginecontrol device. Thus, equation (16) describes a stationary equilibriumstate between the variable force F_(p)(p₃,p₄,s_(acr)) at the wastegateplate, the actuator position s_(acr) and the control pressurep_(acr)(p₀,p_(vac),u_(wg)), which can be affected directly by thecontrol u_(wg).

In systems that do not measure the actuator position, the task ofpre-control of the wastegate for setting the desired charging pressurecan be formulated as follows: For currently occurring pressures p₃upstream from the turbine and p₄ downstream from the turbine, thewastegate control u_(wg) may be selected so the control pressurep_(acr,sp) compensates all other moments acting on the wastegate leverexactly in the nominal wastegate actuator position s_(acr,sp) necessaryfor setting the desired charging pressure. The following applies:

$\begin{matrix}{\mspace{76mu} {{u_{wg} = {f\left( p_{{acr},{sp}} \right)}}{p_{{acr},{sp}} = {p_{0} + {{F_{p}\left( {p_{3},p_{4},s_{{acr},{sp}}} \right)} \cdot \frac{I_{wg}}{A_{acr} \cdot I_{acr}}} + {\frac{{k \cdot s_{{acr},{sp}}} + F_{{spr},a}}{A_{acr}}.}}}}} & (17)\end{matrix}$

This equation (17) cannot be solved directly according to the nominalwastegate or actuator position. Each wastegate pre-control is anapproximation of the function described with equation (17),independently of whether it is described analytically in the enginecontrol device or approximated with characteristic diagrams over severalinput parameters.

The nominal control pressure p_(acr,sp) may be stored as a wastegatepre-control in a memory, the essential inputs of which are the nominalvalues derived from the desired charging pressure for the pressureupstream from the turbine and the mass flow through the wastegate. Theparameters of actuator position and force at the wastegate plate,crucial for a physical description, are not typically used.

Starting from this point, however, the teachings of the presentdisclosure provide an improved method for controlling the actuator ofthe wastegate of an exhaust gas turbocharger of a motor vehicle. In someembodiments, a wastegate model may be used directly or inverted as analgorithm for controlling the turbocharger, as is explained in greaterdetail in the following

A wastegate model or forward model is from this point on one which isdetermined from a known position S_(acr) of the wastegate actuator usingpressures and temperatures of the exhaust gas mass flow m_(wg) flowingthrough the wastegate, assumed to be known, and the force F_(p) actingon the wastegate plate due to the pressure difference at the wastegateplate.

The model describes the wastegate as a system of two throttle points,connected in series, through which in the stationary state of the samethe exhaust gas mass flow flows. This is shown in FIG. 4, which shows adiagrammatic representation of the wastegate as a series connection oftwo throttle points.

FIG. 4 illustrates a constant borehole surface A_(B) and an annularsurface A_(R) of the wastegate, which depends on the actuator positions_(acr). The wastegate mass flow {dot over (m)}_(wg) is the same forboth throttle points and flows first through the borehole surface A_(B)and then through the annular surface A_(R) of the wastegate. An exhaustgas manifold pressure p₃ and an exhaust gas manifold temperature T₃exist upstream from the wastegate. An exhaust gas pressure p₄ which isless than p₃ and an exhaust gas temperature T₄ exist downstream from thewastegate.

Between the borehole surface and the annular surface is a temperaturereferred to hereinafter as the internal wastegate temperature T_(wg).Since the temperature of the gas, when throttled, changes only verylittle, it is assumed hereinafter that the exhaust gas manifoldtemperature T₃ also exists between the borehole surface and the annularsurface.

The pressure drop from p₃ to p₄, which can be measured over the whole ofthe wastegate, is distributed over the two throttle points, depending onthe actuator position. Between the borehole surface and the annularsurface, a pressure therefore exists which is referred to hereinafter asinternal wastegate pressure p_(wg), for which the following relationshipapplies:

p ₃ >p _(wg) >p ₄.

As a simplification, it is assumed that this internal wastegate pressurep_(wg) acts uniformly over the whole of the side of the wastegate plate3, facing the turbine housing with the surface A_(B). Furthermore, it isassumed that the pressure p4, downstream from the turbine, actsuniformly on the whole of the other side of the wastegate plate 3 withthe surface A_(B). The force F_(p), introduced in FIG. 2 and acting onthe wastegate plate due to the pressure difference at the wastegateplate, can thus be described as

$\begin{matrix}\begin{matrix}{F_{p} = {A_{B} \cdot \left( {p_{wg} - p_{4}} \right)}} \\{= {\frac{\pi}{4} \cdot D_{wg}^{2} \cdot \left( {p_{wg} - p_{4}} \right)}}\end{matrix} & (18)\end{matrix}$

A gas mass flow {dot over (m)} through a throttle generally is describedwith the throttle equation

$\begin{matrix}{\overset{.}{m} = {A \cdot s_{sp} \cdot \sqrt{\frac{2 \cdot \kappa}{\left( {\kappa - 1} \right) \cdot R \cdot T_{up}}} \cdot {\Psi \left( \frac{p_{down}}{p_{up}} \right)}}} & (19)\end{matrix}$

with T_(up)=the temperature upstream from the throttle point, p_(up)=thepressure upstream from throttle point, p_(down)=the pressure downstreamfrom throttle point, κ=the isoentropic exponent, R=c_(p)−c_(v)=thespecific gas constant, c_(p)=the specific heat capacity of the gas atconstant pressure and c_(v)=the specific heat capacity of the gas atconstant volume.

The following generally applies for the pressure relationship at thethrottle point:

$\begin{matrix}{\Pi = \frac{p_{down}}{p_{up}}} & (20)\end{matrix}$

wherein p_(down) is the pressure downstream from throttle point andp_(up) the pressure upstream from the throttle point.

Moreover, the following is the relationship for the flow coefficients atthe throttle point for Π<0.53, that is, a supercritical pressurerelationship

$\begin{matrix}{{\Psi (\Pi)} = \left\{ \begin{matrix}{\left( \frac{2}{\kappa + 1} \right)^{\frac{1}{\kappa + 1}}\sqrt{\frac{\kappa - 1}{\kappa + 1}}} \\\sqrt{\left( \frac{p_{down}}{p_{up}} \right)^{\frac{2}{x}} - \left( \frac{p_{down}}{p_{up}} \right)^{\frac{x + 1}{x}}}\end{matrix} \right.} & (21)\end{matrix}$

Applied to the constant borehole surface, the throttle equationdescribes the wastegate mass flow {dot over (m)}_(wg) as

$\begin{matrix}{{{\overset{.}{m}}_{wg} = {A_{B} \cdot p_{3} \cdot \sqrt{\frac{2 \cdot \kappa}{\left( {\kappa - 1} \right) \cdot R \cdot T_{3}}} \cdot {\Psi \left( \frac{1}{\Pi_{B}} \right)}}},} & (22)\end{matrix}$

wherein the following relationship applies for the ratio of pressureupstream from the borehole surface to the pressure downstream from theborehole surface:

$\Pi_{B} = {\frac{p_{3}}{p_{wg}} > 1.}$

Applied to the wastegate position-dependent annular surface, thethrottle equation describes the wastegate mass flow {dot over (m)}_(wg)as:

$\begin{matrix}{\mspace{76mu} {{{\overset{.}{m}}_{wg} = {{A_{B}\left( s_{acr} \right)} \cdot p_{wg} \cdot \sqrt{\frac{2 \cdot \kappa}{\left( {\kappa - 1} \right) \cdot R \cdot T_{3}}} \cdot {\Psi \left( \Pi_{R} \right)}}},{{{with}\text{:}\mspace{14mu} \Pi_{R}} = {{\frac{p_{4}}{p_{wg}} < 1} = {{the}\mspace{14mu} {ratio}\mspace{14mu} {of}\mspace{14mu} {the}\mspace{14mu} {pressure}\mspace{14mu} {up}\mspace{14mu} {to}\mspace{14mu} {upstream}\mspace{14mu} {from}\mspace{20mu} {the}\mspace{14mu} {annular}\mspace{14mu} {surface}\mspace{14mu} {to}\mspace{14mu} {that}\mspace{14mu} {upstream}\mspace{14mu} {from}\mspace{14mu} {the}\mspace{14mu} {annular}\mspace{14mu} {{surface}.}}}}}} & (23)\end{matrix}$

The equations (22) and (23) describe the same wastegate mass flow {dotover (m)}_(wg) and can be regarded as equivalent:

$\begin{matrix}{{\overset{.}{m}}_{wg} = {{A_{B} \cdot p_{3} \cdot \sqrt{\frac{2 \cdot \kappa}{\left( {\kappa - 1} \right) \cdot R \cdot T_{3}}} \cdot {\Psi \left( \frac{1}{\Pi_{B}} \right)}} = {{A_{B}\left( s_{acr} \right)} \cdot p_{wg} \cdot \sqrt{\frac{2 \cdot \kappa}{\left( {\kappa - 1} \right) \cdot R \cdot T_{3}}} \cdot {\Psi \left( \Pi_{R} \right)}}}} & (24)\end{matrix}$

After both sides are divided by the root, the relationship between thesurfaces and pressures at the wastegate follows therefrom:

$\begin{matrix}{{A_{B} \cdot p_{3} \cdot \left( \frac{1}{\Pi_{B}} \right)} = {{A_{s}\left( s_{acr} \right)} \cdot p_{wg} \cdot {{\Psi \left( \Pi_{R} \right)}.}}} & (25)\end{matrix}$

Using equations (11)-(13), the wastegate surface ratio is defined as

$\begin{matrix}{{Q_{A}\left( s_{acr} \right)} = {\frac{A_{E}\left( s_{acr} \right)}{A_{B}} = {\frac{\pi \cdot D_{wg} \cdot \frac{I_{wg}}{I_{acr}} \cdot s_{acr}}{\frac{\pi}{4} \cdot D_{wg}^{2}} = {\frac{4 \cdot I_{wg}}{I_{acr} \cdot D_{wg}} \cdot {s_{acr}.}}}}} & (26)\end{matrix}$

By dividing by A_(B)·P_(wg) and by substituting according to equations(22) and (26), the following results from equation (25):

$\begin{matrix}{{{\frac{p_{3}}{p_{wg}} \cdot {\Psi \left( \frac{1}{\Pi_{B}} \right)}} = {\frac{A_{B}\left( s_{acr} \right)}{A_{B}} \cdot {\Psi \left( \Pi_{B} \right)}}}{{\Pi_{B} \cdot {\Psi \left( \frac{1}{\Pi_{B}} \right)}} = {{Q_{A}\left( s_{acr} \right)} \cdot {\Psi \left( \Pi_{R} \right)}}}} & (27)\end{matrix}$

The left side of the equation (27) is a function solely of the pressurerelationship at the borehole surface Π_(B). Substitute functionsX(Π_(B)) and Φ(Π_(B)) are defined for this term:

$\begin{matrix}{{{X\left( \Pi_{B} \right)} = {\Psi \left( \frac{1}{\Pi_{B}} \right)}}{{\Phi \left( \Pi_{B} \right)} = {{\Pi_{B} \cdot {X\left( \Pi_{R} \right)}} = {\Pi_{B} \cdot {{\Psi \left( \frac{1}{\Pi_{B}} \right)}.}}}}} & (28)\end{matrix}$

Using the substitute function Φ(Π_(B)), equation (27) assumes thefollowing form:

Φ(Π_(B))=Q _(A)(s _(acr))·Ψ(Π_(R)),  (29)

The left side of the equation (29) is a function solely of the pressurerelationship at the borehole surface. The right side of the equation(29) is for a particular actuator position s_(acr), that is, for aparticular value of the surface ratio Q_(A)(s_(acr)) as a parameter, afunction solely of the pressure relationship at the annular surface.Nevertheless, both sides can be portrayed as functions of the twopressure relationships, each of the functions being constant over apressure relationship.

The coordinates [Π_(R),Π_(B)] of the intersection of the two surfaces,shown in FIGS. 7 and 8, are the solutions of equation (27) forQ_(A)(s_(acr))=1. Analogously, the coordinates [Π_(R),Π_(B)] of theintersection of the Φ(Π_(R),Π_(B)) surface, shown in FIG. 7, with theΨ(Π_(R),Π_(B)) surface shown in FIG. 8 and scaled by an arbitrarysurface ratio Q_(A)(s_(acr))>0, are the solutions of the equation (27)for this arbitrary surface factor.

Therefore, the coordinates [Π_(R),Π_(B)] of the intersection, so foundand dependent exclusively on the surface ratio Q_(A)(s_(acr)), describeall combinations of pressure relationships at the borehole surface andthe annular surface of the wastegate possible for this given actuatorposition s_(acr).

From the definition of the pressure relationships at the boreholesurface and the annular surface of equations (22) and (23), it followsthat:

$\begin{matrix}{\frac{\Pi_{B}}{\Pi_{R}} = {\frac{\frac{p_{3}}{p_{wg}}}{\frac{p_{4}}{p_{wg}}} = {\frac{p_{3}}{p_{4}} = {{\tan (\alpha)}..}}}} & (30)\end{matrix}$

With that, for a certain stationary combination of pressures p3 upstreamfrom the turbine and p4 downstream from the turbine, the ratio of allpossible combinations of the pressure relationships at the boreholesurface and the annular surface of the wastegate is constant, that is,all possible combinations of the pressure relationships form a straightline g, which passes through the coordinate origin and is drawn in FIG.9, in the [Π_(R),Π_(B)] plane, which is inclined against the

Π_(R) axis by the angle

$\alpha = {{\arctan \left( \frac{\Pi_{B}}{\Pi_{R}} \right)}.}$

With that, the coordinates [Π_(R),Π_(B)] of the straight line, which areso found and dependent exclusively on the pressure relationship

$\frac{p_{3}}{p_{4}},$

describe all possible combinations of the pressure relationships at theborehole surface and the annular surface of the wastegate possible forthis given turbine pressure relationship

$\frac{p_{3}}{p_{4}}.$

The pressure downstream from the wastegate is always smaller than thepressure upstream from the wastegate, that is, p₃>p₄. From this itfollows that

$\begin{matrix}{\frac{p_{3}}{p_{4}} = \left. {{\tan (\alpha)} > 1}\Rightarrow{\alpha > {45{{^\circ}.}}} \right.} & (31)\end{matrix}$

FIG. 9 shows a graphic solution of the equation (27) for a surface ratioQ_(A)(s_(acr))<1, namely the intersection S1 of the left side of theequation, which is illustrated by the K1 formation (see also FIG. 8),and of the right side of the equation, which is illustrated by the K2formation (see also FIG. 7). The projection of the intersection S1 ontothe [Π_(R),Π_(B)] plane, which is illustrated by the broken line S2, isthe quantity of all combinations of pressure relationships at theborehole surface and the annular surface possible for thisQ_(A)(s_(acr)).

With that, the straight line has always exactly one point ofintersection G=[Π_(R),Π_(B)] with the projection of the intersectiononto the Π[_(R),Π_(B)] plane, that is, the coordinates of the point ofintersection G=[Π_(R),Π_(B)] are the only solution of the equationsystem, which is formed from equations (27) and (30).

$\begin{matrix}{{{\Pi_{s} \cdot {\Psi \left( \frac{1}{\Pi_{R}} \right)}} = {{Q_{A}\left( s_{acr} \right)} \cdot {\Psi \left( \Pi_{R} \right)}}}{\Pi_{R} = {\frac{p_{3}}{p_{4}} \cdot \Pi_{R}}}} & (32)\end{matrix}$

and the equation with Π_(R) as single variable obtained therefrom by theelimination of Π_(B).

$\begin{matrix}{{\frac{p_{3}}{p_{4}} \cdot \Pi_{R} \cdot {\Psi \left( \frac{p_{4}}{p_{3} \cdot \Pi_{R}} \right)}} = {{Q_{A}\left( s_{acr} \right)} \cdot {\Psi \left( \Pi_{R} \right)}}} & (33)\end{matrix}$

This equation (33) can thus be solved numerically for any combinationsof

$\frac{p_{3}}{p_{4}} > 1$

Q_(A)(s_(acr))>0. This solution, with the successful modelingsimplification of the wastegate as a series connection of two throttlepoints and the disregard of the pulsation of the exhaust gas mass flow,is valid globally for all wastegate turbochargers in all stationaryoperating points.

The stationary pressure relationships, so determined over the annularsurface of the wastegate

${\Pi_{R,{acr}}\left( {\frac{p_{3}}{p_{4}},Q_{A}} \right)},$

are filed as a constant characteristic diagram in the engine controldevice. FIG. 10 shows the global stationary pressure relationship overthe annular surface of the wastegate

${\Pi_{R,{acr}}\left( {\frac{p_{3}}{p_{4}},Q_{A}} \right)}.$

To sum up, at the running time in the engine control device, the exhaustgas mass flow can be calculated by the wastegate {dot over (m)}_(wg)from the constant wastegate borehole diameter D_(wg), the constantwastegate lever lengths l_(wg),l_(acr), the constant isoentropicexponent κ, the constant specific gas constant R of the exhaust gas, thecurrent position of the wastegate actuator S_(acr), the current pressurep₃ upstream from the turbine, the current pressure p₄ downstream fromthe turbine and the current temperature T₃ upstream from the turbine.

The borehole surface of the wastegate is constantly calculated for alloperating points from equation (11)

$\begin{matrix}{A_{B} = {{\frac{\pi}{4} \cdot D_{wg}^{2}}..}} & (34)\end{matrix}$

From the current position of the wastegate actuator s_(acr), the currentannular surface follows according to equations (12) and (13)

$\begin{matrix}{A_{R} = {{\pi \cdot D_{wg} \cdot s_{acr}} = {\pi \cdot D_{sp} \cdot \frac{I_{wg}}{I_{acr}} \cdot {s_{acr}.}}}} & (35)\end{matrix}$

The wastegate surface ratio follows from equation (26)

$\begin{matrix}{Q_{A} = {\frac{A_{R}}{A_{B}}.}} & (36)\end{matrix}$

The stationary pressure relationship over the annular surface of thewastegate Π_(R) is read from the stored characteristic diagram

$\begin{matrix}{\Pi_{R} = {{\Pi_{R,{acr}}\left( {\frac{p_{3}}{p_{4}},Q_{A}} \right)}.}} & (37)\end{matrix}$

According to equation (23), the internal wastegate pressure p_(wg) is:

$\begin{matrix}{p_{wg} = {\frac{p_{4}}{\Pi_{R}}.}} & (38)\end{matrix}$

According to equation (18) the force on the wastegate plate resultingtherefrom is

$\begin{matrix}{F_{p} = {\frac{\pi}{4} \cdot D_{wg}^{2} \cdot {\left( {p_{wg} - p_{4}} \right).}}} & (39)\end{matrix}$

According to equation (23), the current waste gas mass flow finally is

$\begin{matrix}{{\overset{.}{m}}_{wg} = {A_{R} \cdot \sqrt{\frac{2 \cdot \kappa}{\left( {\kappa - 1} \right) \cdot R \cdot T_{3}}} \cdot \rho_{wg} \cdot {{\Psi \left( \Pi_{R} \right)}.}}} & (40)\end{matrix}$

The wastegate forward model in the engine control device may be used forturbochargers, which are equipped with both variable turbine geometry(VTG), as the main actuator, as well as with a wastegate as an auxiliaryactuator. For VTG turbochargers without wastegate, all the exhaust gasmass flow of the engine is passed through the turbine. With that, theexhaust gas mass flow, available at the turbine, is known for thecalculation of the VTG control. For VTG turbochargers with an additionalwastegate, it is possible to calculate according to equation (2) theportion of the exhaust gas mass flow of the engine, which is availablefor a selected actuator position s_(acr) at the turbine:

{dot over (m)} _(tur) ={dot over (m)} _(eng) −{dot over (m)} _(wg)(s_(acr))  (41).

The further calculation of the VTG control can then be carried out asfor VTG turbochargers without an additional wastegate.

A model is referred to in the following as an inverse wastegate model(backwards model), which, using pressures and temperatures from anominal exhaust gas mass flow through the wastegate {dot over(m)}_(wg,sp), assumed to be known, determines the nominal position ofthe wastegate actuator s_(acr,sp) and the nominal force on the wastegateplate F_(p,sp) required for the realization of the wastegate {dot over(m)}_(wg,sp).

For typical wastegate turbochargers without a variable turbine geometry,according to equation (2) and starting out from the current exhaust gasmass flow through the engine {dot over (m)}_(eng) and the nominalexhaust gas mass flow through the turbine m_(tur,sp) resulting from thedriver's request, a nominal exhaust gas mass flow through the wastegate{dot over (m)}_(wg,sp) is calculated:

{dot over (m)} _(wg,sp) ={dot over (m)} _(eng) −{dot over (m)}_(tur,sp)  (42).

The throttle equation (23) for the annular surface is analogously validfor nominal values:

$\begin{matrix}{{\overset{.}{m}}_{{wg},{sp}} = {{A_{R,{sp}}\left( s_{{acr},{sp}} \right)} \cdot p_{{wg},{sp}} \cdot \sqrt{\frac{2 \cdot \kappa}{\left( {\kappa - 1} \right) \cdot R \cdot T_{3}}} \cdot {{\Psi \left( \Pi_{R,{sp}} \right)}.}}} & (43)\end{matrix}$

The nominal value of the internal wastegate pressure is according toequation (23), the nominal annular surface replaced according toequation (26):

$\begin{matrix}{{\overset{.}{m}}_{{wg},{sp}} = {{Q_{A,{sp}}\left( s_{{acr},{sp}} \right)} \cdot A_{B} \cdot \frac{p_{4}}{\Pi_{R,{sp}}} \cdot \sqrt{\frac{2 \cdot \kappa}{\left( {\kappa - 1} \right) \cdot R \cdot T_{3}}} \cdot {{\Psi \left( \Pi_{R,{sp}} \right)}.}}} & (44)\end{matrix}$

Rearranging results in the following:

$\begin{matrix}{{{Q_{A,{sp}}\left( s_{{acr},{sp}} \right)} \cdot \frac{\Psi \left( \Pi_{R,{sp}} \right)}{\Pi_{R,{sp}}}} = {\frac{{\overset{.}{m}}_{{wg},{sp}}}{A_{B} \cdot p_{4} \cdot \sqrt{\frac{2 \cdot \kappa}{\left( {\kappa - 1} \right) \cdot R \cdot T_{3}}}} = {W_{sp}.}}} & (45)\end{matrix}$

The equation (45) is to be understood as implying that, for a requirednominal exhaust gas mass flow through the wastegate {dot over(m)}_(wg,sp) at a known pressure p₄ downstream from the turbine and at aknown temperature T₃ upstream from the turbine, a combination ofwastegate surface ratio Q_(A,sp)(s_(acr,sp)) and pressure relationshipat the annular surface of the wastegate Π_(R,sp), bringing about thismass flow, is to be found. The parameter, defined in equation (45) isreferred to as nominal mass flow factor W_(sp).

The stationary pressure relationship over the annular surface of thewastegate is filed as characteristic diagram over the turbine pressurerelationship

$\frac{p_{3}}{p_{4}}$

and the wastegate surface ratio Q_(A) (see equation (37)). For eachpoint of this characteristic diagram, the mass flow factor can becalculated according to equations (45) and (21) as

$W = {Q_{A} \cdot \frac{\Psi \left( {\Pi_{R}\left( {\frac{p_{3}}{p_{4}},Q_{A}} \right)} \right)}{\Pi_{R}\left( {\frac{p_{3}}{p_{4}},Q_{A}} \right)}}$

and filed in an equally large characteristic diagram

${W\left( {\frac{p_{3}}{p_{4}},Q_{A}} \right)}.$

This mass flow factor, like the stationary pressure relationship overthe annular surface of the wastegate with the simplification globallymade for all wastegate turbochargers, is also valid at all stationaryoperating points.

FIG. 11 illustrates the characteristic diagram of the mass flow factor

${W\left( {\frac{p_{3}}{p_{4}},Q_{A}} \right)}.$

This characteristic diagram

$W\left( {\frac{p_{3}}{p_{4}},Q_{A}} \right)$

is strictly monotonic and can be inverted off-line according to Q_(A)into a nominal surface ratio characteristic diagram

$\left| {Q\left( {\frac{p_{3}}{p_{4}},W} \right)} \right.$

and filed in the engine control device. This nominal surface ratiocharacteristic diagram, with the simplification made globally for allwastegate turbochargers, is also valid in all stationary operatingpoints. The nominal surface ratio Q_(A,sp), realizing a nominal value ofthe mass flow factor W_(sp), can be selected from this characteristicdiagram for the current turbine pressure relationship

$\frac{p_{3}}{p_{4}}$

for said nominal value of the mass flow factor W_(sp).

$\begin{matrix}{Q_{A,{sp}} = {{Q\left( {\frac{p_{3}}{p_{4}},W_{sp}} \right)}.}} & (46)\end{matrix}$

The nominal actuator position can then be determined from the invertedequation (26)

$\begin{matrix}{s_{{acr},{sp}} = {Q_{A,{sp}} \cdot {\frac{I_{acr} \cdot D_{wg}}{4 \cdot I_{wg}}.}}} & (47)\end{matrix}$

By using equations (37) to (39) for the nominal area ratio Q_(A,sp), thenominal force on the wastegate plate F_(p,sp), corresponding to this, isdetermined.

To summarize, from a nominal exhaust gas mass flow {dot over(m)}_(wg,sp) through the wastegate, the nominal position s_(acr,sp) ofthe wastegate actuator and the nominal force F_(p,sp) on the wastegateplate required for the implementation of said nominal exhaust gas massflow can be determined at the running time in the engine control device,from the constant wastegate borehole diameter D_(wg), the constantwastegate lever length l_(wg),l_(acr), the constant isoentropic exponentκ, the constant specific gas constant R of the exhaust gas, the currentpressure p₃ upstream from the turbine, the current pressure p₄downstream from the turbine and the current temperature T₃ upstream fromthe turbine.

The nominal mass flow factor is determined according to equation (45)from the nominal wastegate exhaust gas mass flow {dot over (m)}_(wg,sp):

$\begin{matrix}{W_{sp} = {\frac{{\overset{.}{m}}_{{wg},{sp}}}{A_{B} \cdot p_{A} \cdot \sqrt{\frac{2 \cdot \kappa}{\left( {\kappa - 1} \right) \cdot R \cdot T_{3}}}}.}} & (48)\end{matrix}$

According to equation (46) the nominal wastegate surface ratio is readfrom the nominal wastegate area ratio characteristic diagram, filed inthe engine control device:

$\begin{matrix}{Q_{A,{sp}} = {{Q\left( {\frac{p_{3}}{p_{4}},W_{sp}} \right)}.}} & (49)\end{matrix}$

The final nominal actuator position s_(acr,sp) is determined fromequation (47)

$\begin{matrix}{s_{{acr},{sp}} = {Q_{A,{sp}} \cdot {\frac{I_{acr} \cdot D_{wg}}{4 \cdot I_{wg}}.}}} & (50)\end{matrix}$

According to equation (37), the nominal pressure relationship over theannular surface of the wastegate Π_(R,sp) is read from the storedcharacteristic diagram:

$\begin{matrix}{\Pi_{s,{sp}} = {\Pi_{A,{acr}}\left( {\frac{p_{3}}{p_{4}},Q_{A,{sp}}} \right)}} & (51)\end{matrix}$

According to equations (38) and (39), the internal nominal wastegatepressure p_(wg,sp) and the nominal force on the wastegate plate F_(p,sp)resulting therefrom are

$\begin{matrix}{p_{{wg},{sp}} = \frac{p_{4}}{\Pi_{A,{sp}}}} & (52) \\{F_{p,{sp}} = {\frac{\pi}{4} \cdot D_{wg}^{2} \cdot \left( {\rho_{{wg},{sp}} - p_{4}} \right)}} & (53)\end{matrix}$

Finally, the nominal actuator pressure p_(acr,sp), required for settingthe desired charging pressure and therefrom the wastegate control u_(wg)is calculated from this nominal value combination s_(acr,sp) andF_(p,sp) according to equation (17) for wastegate turbochargers with apneumatic wastegate actuator without measuring the actuator position:

$\begin{matrix}{{p_{{acr},{sp}} = {p_{0} + {{F_{p}\left( {p_{3},p_{4},s_{{acr},{sp}}} \right)} \cdot \frac{I_{wg}}{A_{acr} \cdot I_{acr}}} + \frac{{k \cdot s_{{acr},{sp}}} + F_{{acr},{sp}}}{A_{acr}}}}\mspace{76mu} {u_{wg} = {f\left( p_{{acr},{sp}} \right)}}} & (54)\end{matrix}$

Alternatively, the calculation chain (48) to (53) can also be used forcontrolling the wastegate turbochargers with measurement of thewastegate actuator position. A wastegate actuator position control,previously based only on the nominal actuator position s_(acr,sp), canbe made more robust there by taking into consideration the additionalnominal force on the wastegate plate F_(p,sp) as a known interferingparameter.

The pre-control of wastegate turbochargers may be improved by employingthe methods taught herein. It may differentiate better between variousoperating states than is possible with precontrol which is notphysically based. With that, the respective best control can becalculated and there is less need for a correction of the precontrol bya boosting pressure controller. Overall, the response behavior of thecombustion engine is improved.

What is claimed is:
 1. A method for controlling an actuator of thewastegate of an exhaust gas turbocharger of a motor vehicle, the methodcomprising: characterizing the wastegate in a model as a seriesconnection of two throttle points; and actuating the wastegate based onthe model.
 2. The method as claimed in claim 1, further comprising:filing a characteristic diagram in a memory of an engine control deviceof the motor vehicle; and determining the nominal relationship ofannular surface to borehole surface of the wastegate as a function ofthe pressure relationships at the wastegate and as a function of anominal mass flow factor.
 3. The method as claimed in claim 2, furthercomprising: calculating the nominal wastegate exhaust gas mass flow at acurrent operating point during the running time of the exhaust gasturbocharger, calculating a nominal mass flow factor associated with thecurrent operating point based on the determined nominal wastegateexhaust gas mass flow; calculating a nominal wastegate-area relationshipassociated with the current operating point based on the filedcharacteristic diagram using the determined nominal mass flow factor;and calculating a nominal position of the actuator to realize a requirednominal wastegate mass flow at the current operating point.
 4. Themethod as claimed in claim 3, further comprising: calculating a nominalforce acting on a wastegate plate of the wastegate; calculating anominal actuator pressure required to set a desired charging pressurebased on the determined nominal position and the determined nominalforce; and calculating a control signal for the actuator based on thenominal actuator pressure calculated.